17 research outputs found
A Phase transition in zero count probability for Stationary Gaussian Processes
We study the probability that a real stationary Gaussian process has at least
zeros in (overcrowding), or at most this number
(undercrowding). We show that if the spectral measure of the process is
supported on , overcrowding probability transitions from exponential
decay to Gaussian decay at , while undercrowding
probability undergoes the reverse transition at .Comment: 17 page
3/2 Firefighters are not enough
The firefighter problem is a monotone dynamic process in graphs that can be
viewed as modeling the use of a limited supply of vaccinations to stop the
spread of an epidemic. In more detail, a fire spreads through a graph, from
burning vertices to their unprotected neighbors. In every round, a small amount
of unburnt vertices can be protected by firefighters. How many firefighters per
turn, on average, are needed to stop the fire from advancing? We prove tight
lower and upper bounds on the amount of firefighters needed to control a fire
in the Cartesian planar grid and in the strong planar grid, resolving two
conjectures of Ng and Raff.Comment: 8 page